Systems and methods for improving calibration of a linear array sensor

ABSTRACT

Systems and methods provide calibration for a linear array sensor. A test pattern having a plurality of lines is used. Comparison between expected and measured spacing between a pair of neighboring lines is used to determine sensor position displacement.

This is a Divisional of application Ser. No. 10/953,321 filed Sept. 30,2004. The entire disclosure of the prior application is herebyincorporated by reference herein in its entirety.

BACKGROUND OF THE INVENTION

1. Field of Invention

This invention relates to systems and methods for improving calibrationof a linear array sensor.

2. Description of Related Art

Linear array sensors have been used in binary applications, such as, forexample, in a bar code reader. In these applications, each sensorelement is used to determine whether light is on or off.

A linear array sensor consists of a series of light sensitive pixels.The pixels are generally characterized by a dark current and a gain. Thedark current gives rise to a charge on the sensor when there is no lightinput. The gain gives the additional charge built up on each pixel perunit of light input.

The dark current and gain of each pixel can be different. When lineararray sensors are used for input scanners, the sensitivity of thesensors is calibrated by measuring the response of each pixel under noillumination and the response of each sensor scanning a uniform strip.These two measurements are used to calculate a gain and an offset forthe each pixel. This calibration is used to derive a linear equation torelate the charge buildup on each pixel, to the reflection from asurface.

SUMMARY OF THE INVENTION

The voltage increase of linear array sensors in response to light inputis linear for small light exposures, but saturates for high lightexposures. Therefore, for some applications, it may be necessary to usea non-linear equation to relate the response of the sensor to the lightexposure.

Some linear array sensors are composed of separate sensor chips buttedtogether. The spacing between neighboring pixels across the sensor chipsmay not be equal to the spacing between the pixels within a sensor chip.In addition, the sensor chip may be slightly rotated with respect to theaxis of the sensor, changing the pixel spacing projected onto the axisof the sensor. Some applications of a linear array sensor require anaccurate knowledge of the spacing between each pixel. One particularapplication is the use of the linear array as a registration sensor in adirect marking printer.

Various exemplary embodiments of the systems and methods according tothis invention provide a method for a spatial calibration of a lineararray sensor, comprising: providing a target that may have one or morestrips; each strip comprising a line target having a plurality of linesof known spacing; obtaining a series of linear array sensor profiles foreach strip at different relative positions of the linear array sensor tothe strips in a direction perpendicular to the direction of the lines;extracting the centers of the lines from the linear array sensorprofile; and determining a displacement associated with each pixel ofthe linear array sensor based on the known spacing and the measuredspacing of the lines.

Various exemplary embodiments of the systems and methods according tothis invention provide a method for gray level calibration of a lineararray sensor in applications where a substrate of nonuniformreflectivity is monitored. The substrate can be a belt or a drum.

In various exemplary embodiments, the substrate is a drum. The drum hasan axis and rotates about the axis. The method comprises: determining afirst set of calibration parameters when the drum is at a first rotationangle about the axis, the first set of calibration parameters associatedwith calibrating the linear array sensor in a direction along the axiswhen the drum is at the first rotation angle about the axis; determininga second set of calibration parameters when the drum is at a secondrotation angle about the axis, the second set of calibration parametersassociated with calibrating the linear array sensor in the directionalong the axis when the drum is at the second rotation angle about theaxis; and storing the first and second sets of calibration parameters.

In various other exemplary embodiments, the substrate is a belt. Thebelt moves in a process direction along a path. The method comprises:determining a first set of calibration parameters when a reference pointof the belt is at a first location along the path, the first set ofcalibration parameters associated with calibrating the linear arraysensor in a direction perpendicular to the path when the belt is at thefirst location; determining a second set of calibration parameters whenthe reference point is at a second location in the path, the second setof calibration parameters associated with calibrating the linear arraysensor in the direction perpendicular to the path when the referencepoint is at the second location; and storing the first and second setsof calibration parameters.

In various exemplary embodiments, more than two sets of parameters areobtained and stored. In such exemplary embodiments, a number of n=C/Ssets of parameters are stored, where C is the circumference of the drumor the length of the belt, and S is a distance over which thecalibration may change.

In various exemplary embodiments, more than two levels of illuminationare provided on the substrate. The corresponding responses at the linearsensor are used for nonlinear gray level calibration.

This and other features and advantages of this invention are describedin, or are apparent from, the following detailed description of variousexemplary embodiments of the systems and methods according to thisinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

Various exemplary embodiments of the systems and methods of thisinvention will be described in detail, with reference to the followingfigures, wherein:

FIG. 1 illustrates an exemplary embodiment of a test pattern, areceiving member and a sensor according to this invention.

FIG. 2 is similar to FIG. 1, except that the sensor is translated in thecross process direction.

FIG. 3 illustrates an exemplary embodiment of a test pattern, a flatsubstrate and a sensor according to this invention.

FIG. 4 is similar to FIG. 3, except that the sensor is translated in thecross process direction.

FIG. 5 illustrates an exemplary embodiment of a test pattern accordingto this invention;

FIG. 6 is a flowchart outlining an exemplary embodiment of a calibrationmethod according to this invention;

FIG. 7 is a flowchart outlining another exemplary embodiment of acalibration method according to this invention; and

FIG. 8 is a functional block diagram of an exemplary embodiment of acalibration system according to this invention.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

Various exemplary embodiments of systems and methods according to thisinvention provide improved calibration for linear array sensors. Invarious exemplary embodiments, the calibration may be performed before alinear array sensor is mounted in a printer. In this exemplaryembodiment, the test pattern is printed on paper or some othersubstrate. In this exemplary embodiment, the linear array sensor may bea full-width array sensor. In various other exemplary embodiments, thecalibration may be performed in the printer on the substrate that thelinear array sensor is imaging.

FIG. 1 illustrates an exemplary embodiment of a test pattern 10, a drum20 and a sensor 30 according to this invention. In the system shown inFIG. 1, the test pattern 10 consists of a set of one or more strips 12,14. Each strip 12, 14 consists of a set of lines 16 oriented in theprocess direction 40 and spaced in the cross process direction 50. Thesensor 30 monitors the test pattern 10 as the drum 20 rotates under thesensor 30. In various exemplary embodiments, the spacing between thelines 16 is known.

In various exemplary embodiments, the spacing of the lines in differentstrips 12 and 14 are the same. In various other exemplary embodiments,the spacing of the lines in different strips 12 and 14 are different.

In various exemplary embodiments, the distances between differentneighboring pairs of the lines 16 are regular or the same. In variousother exemplary embodiments, the distances between different neighboringpairs of the lines 16 are different or random.

In various exemplary embodiments, the calibration is made by monitoringthe test pattern 10 when the linear array sensor 30 is in a fixedposition. In various other exemplary embodiments, the calibration ismade by obtaining a plurality of measurements with the linear arraysensor 30 moved in the cross process direction 40, which isperpendicular to the direction of the plurality of lines.

FIG. 2 illustrates the same exemplary embodiment shown in FIG. 1, butwith the sensor 30 translated in the cross process direction 50.

The sensor 30 can be calibrated before being used in a printer. FIG. 3illustrates an exemplary embodiment of a test pattern 10, a substrate,such as paper 21, with the test pattern 10, and a sensor 30. In thesystem shown in FIG. 3, the test pattern 10 consists of a set of one ormore strips 12, 14. Each strip 12, 14 consists of a set of lines 16oriented in the process direction 40 and spaced in the cross processdirection 50. The sensor 30 can monitor the test pattern 10 as thesubstrate 21 moves under the sensor 30. Alternatively, the sensor 30 canmonitor the test pattern 10 as the sensor 30 is moved over the testpattern 10.

FIG. 4 illustrates the same exemplary embodiment shown in FIG. 3, butwith the sensor 30 translated in the cross process direction 50 which isperpendicular to the orientation of the test pattern lines 16.

In various exemplary embodiments, measurements are obtained for thelinear array pixel response profile for a dark level and for a singleexposure level. In such exemplary embodiments, a gain and an offset foreach pixel of the linear array may be established.

In various other exemplary embodiments, measurements are obtained forthe linear array pixel response profile for a dark level and for morethan one light exposure levels. In such exemplary embodiments, anon-linear relationship may be established between each pixel and thelight exposure.

In various exemplary embodiments, to establish the non-linearrelationship for each pixel in a linear sensor before it is mounted in aprinter (calibration-on-paper technique), the linear array pixelresponse profile may be mapped by varying the illumination level andmonitoring the response to a uniform patch of high reflectivity. Analternative for the calibration-on-paper technique is to monitor aplurality of uniform surfaces, each having a different reflectivity.

In various exemplary embodiments, for establishing the non-linearrelationship for each pixel of a linear sensor after the sensor ismounted on the drum (calibration-on-drum technique), the linear arraypixel response profile may be mapped by varying the illumination whilethe sensor monitors a fixed location on the drum surface.

In various exemplary embodiments, to establish the non-linearrelationship for each pixel of a linear sensor, the sensor response tomore than one levels of illumination intensity is measured. Typically,the response is linear for low levels of illumination. Also typically,at higher illumination levels, the sensitivity of sensor responsedecreases until it starts to saturate. For calibration purposes, varioustechniques may be used to relate the sensor response to the localreflectivity of the surface. In one exemplary embodiment, theintermediate responses may be interpolated between the measuredresponses for each sensor pixel. In other various exemplary embodiments,a function may be fit to each pixel response as a function ofillumination intensity. In various exemplary embodiments, a quadraticpolynomial function is used to fit the sensor responses. In variousother exemplary embodiments, an exponential function is used.

In various exemplary embodiments, a table of coefficients of the fittingfunction is obtained. The coefficients are used to expeditiously converta raw sensor measurement into a calibrated sensor measurement whenneeded.

Various exemplary embodiments of the systems and methods according tothis invention provide for marking device receiving member signaturecalibration. In one exemplary embodiment, the variations of the drum inreflection may have an x-axis signature component, as a function of theaxis of the drum (the x-direction), and a rotational signaturecomponent, as a function of the rotation angle θ of the drum. In anotherexemplary embodiment, the variations of the belt in reflection may havean x-axis signature component, as a function of the cross processposition on the belt (the x-direction), and a process directionsignature component, as a function of the cross process position on thebelt.

In the exemplary embodiment in which the receiving member is a drum, thevariation of the reflection along the axis of the drum for a fixed θ maybe confounded with the linear sensor pixel profile. In various exemplaryembodiments, for the calibration-on-drum technique, the calibration ofthe sensor response may automatically calibrate for the x-axis drumsignature.

When the drum reflection variation along the x-axis is separable fromthe drum reflection variation as a function of θ, the drum reflectionmay be expressed as a product of the x-axis component and the rotationalcomponent:r(x, θ)=r_(x)(x)r_(θ)(θ),   (1)where r is the reflectivity of the drum surface.

In various exemplary embodiments, the reflection variation as a functionof 0 (the rotational signature) may be measured by measuring a linearsensor profile and determining the average response of all the pixels.The average sensor response is monitored as a function of rotation anglefor a fixed illumination. The average sensor response is used as a scalefactor for the calibration. During operation of the sensor, the rotationangle is monitored using a drum encoder, with an appropriate scalefactor. The sensor profile is multiplied by the scale factor precedingfurther processing of the linear sensor image.

When the drum reflection variations in the x-axis direction are notseparable from the rotational signature, the rotational signature maydepend on a geometric parameter, such as the runout of the drum, or avariation in the property of the drum surface. In various exemplaryembodiments, a separate calibration is performed at each rotationalangle of the drum. A relationship may be established between drumrotational angles and the period of variations. During operation of thesensor, the rotation angle is monitored using a drum encoder, with anappropriate calibration table.

In various exemplary embodiments using a drum, n sets of calibrationparameters are obtained and stored, where n is an integer greater thantwo and less than n=C/S, C being a circumference of the drum, and Sbeing a distance on a surface of the drum corresponding to an angulardifference between two rotation angles over which the calibration canchange significantly.

In the exemplary embodiment in which the receiving member is a belt, thevariation of the reflection in the cross process direction for a fixedprocess direction position may be confounded with the linear sensorpixel profile. In various exemplary embodiments, the calibration of thesensor response may automatically calibrate for the cross processdirection belt signature. When the belt reflection variation in thecross process direction is separable from the belt reflection variationas a function of the process direction position, the belt response maybe calibrated in a way similar to that expressed in Equation (1), exceptthat the rotational component r_(θ)(θ) of the drum needs to be replacedby a corresponding process direction component of the belt.

In various exemplary embodiments using a belt, n sets of calibrationparameters are obtained and stored, where n is an integer greater thantwo and less than n=C/S, C being the length of the belt, and S being adistance on the belt between two process direction positions over whichthe calibration can change significantly.

Various exemplary embodiments of the systems and methods according tothis invention provide sensor pixel position calibration. In variousexemplary embodiments, a sensor pixel position calibration may be madeby measuring a random line target on a receiving member. The receivingmember may include an image recording medium, a photoreceptor drum, anintermediate belt, or the like.

FIG. 5 illustrates an exemplary embodiment of the test pattern 120 shownin FIG. 1. As shown in FIG. 5, the test pattern 120 may include aplurality of lines 121 on a receiving member. In various exemplaryembodiments, the lines are substantially parallel to each other. Eachline extends in the process direction 210 of the printer.

In various exemplary embodiments, for the calibration-on-papertechnique, the line spacing 122 in the test pattern is known. Forexample, the line spacing has been accurately measured using anothertechnique. Or, the device that prints the lines is know a priori toaccurately position the lines. In various exemplary embodiments, for thecalibration-on-drum technique, the test pattern is printed on the drumitself.

In various exemplary embodiments, a calibration is needed for a directmarking print engine. The ink ejected from a direct marking print headcan have some misdirectionality in the cross process position that isdifferent for each nozzle, but is constant over the time it takes towrite the test pattern. In various exemplary embodiments, thismisdirection may be mitigated by printing test patterns containing setsof lines printed by a same nozzle. In this configuration, the separationof dashes printed only with the same nozzle is compared and analyzed incalibration. In various exemplary embodiments, the separation iscontrolled by printing the test pattern with multiple passes, and thenmeasuring the displacement between passes with an x-axis encoder. Theseparation between test pattern lines can be derived from the x-axisencoder response.

The spatial calibration is performed by analyzing one or more linearsensor response profiles taken when the drum is positioned under thesensor. The presence of printed lines changes sensor response. Inparticular, the presence of ink on the drum can either decrease orincrease the response of sensors, depending on the relative colors ofthe ink and the drum and the texture of the ink and the drum. For theease of discussion, it is assumed that the presence of ink decreasessensor response. However, it should be appreciated that the discussionbelow also applies when the presence of ink increases sensor response.

In various exemplary embodiments, a cross section 230 of sensor responseis used for calibration. As shown in FIG. 5, the cross section 230 ofthe sensor response is a collection of profiles through the lines ordashes 121 in the test pattern 120. A profile includes sensor responsealong the cross process direction 220 at a particular process directionlocation. In various exemplary embodiments, the cross section 230 is acollection of profiles through all the dashes 121 in the test pattern120. In various other exemplary embodiments, the cross section 230 is acollection of profiles through the dashes 121 near an area wherecalibration is required.

In a response profile of a cross section 230 of sensor response, sensorresponse maxima occur at locations corresponding to positions wheredashes 121 do not exist, such as the gaps 123 between dashes 121. On theother hand, sensor response minima occur in the response profile atpositions corresponding to locations where dashes 121 are printed. Thepositions of the minima are used to obtain the locations of thecorresponding dashes 121.

In various exemplary embodiments, the centers 124 of the lines or dashesmay be determined based on the cross section of sensor response, usingthe minima in the response profile. The determination may be achieved byany existing or later developed techniques. In various exemplaryembodiments, the center 124 of a .dash line is determined based on aninterpretation of the response data near the dash line, a mid-point ofthe line edges of a detected dash line, a non-linear least squares fit,or a multi-dimension vector under the Radar theory.

In various exemplary embodiments the test pattern 120 includes S strips(see FIGS. 1-4). Each strip can be measured one or more times, with thesensor in a different cross process position. Each strip can contain Nlines. N can be different for each strip, but for simplicity, we assumeit is equal for all strips.

In various exemplary embodiments, the spacing 122 between neighboringlines 121 which is known is used. Let P_(i, j+1)−P_(i, j) be thedifference in the positions of the line indexed by j+1 and the lineindexed by j for measurement number i. Note that more than onemeasurement can be taken per strip for the linear sensor in a differentcross process position. The difference between the measured and expectedspacing may be expressed as:Δx_(p) _(i, j+1) −Δx_(p) _(i, j)=(p_(i, j+1)−p_(i, j))−(x_(i, j+1)−x_(i, j)),   (2)where the first term on the right hand side of the equation is the knownspacing 122.

In one exemplary embodiment, sufficient measurements are made so that Δxis sampled as finely as the spacing between pixel elements on the lineararray sensor. In another exemplary embodiment, less measurements aremade and an interpolation technique is used to determine the offsets ofthe pixel elements that have not sampled the position of a line from theones that have sampled the position of a line.

A sparse matrix equation may be established by obtaining Equation (2)for a plurality of neighboring line pairs in the test pattern 120. Inaddition, an overdetermined matrix equation may be obtained byestablishing the Equation (2) for a plurality of measurements. Invarious exemplary embodiments, the Moore-Penrose technique or anotherequivalent technique may be used to solve this matrix equations.

In various exemplary embodiments, the solution obtained from the matrixequations is used as a displacement associated with each pixel elementof the sensor. This displacement may be used to fine tune thecalculation of the line centers 124. For example, when a line center 124is found to be centered under a particular sensor, and this center 124was determined as being displaced 10 μm, then the actual line centerwill be measured as 10 μm different from what is assumed based onequally spaced center positions.

FIG. 6 is a flowchart outlining an exemplary embodiment of a calibrationmethod using the calibration-on-paper technique. As shown in FIG. 6, themethod starts at step S100 and continues to step S110, where lineararray sensor responses to illumination levels are obtained.

Next, in step S120, a function is fit to the sensor responses obtained.In various exemplary embodiments, a linear function is fit to extractparameters such as the dark current and the gain. In various otherexemplary embodiments, a non-linear function is fit to extractcoefficients, including parameters that indicate saturation of theresponse.

Then, in step S130, measurements are obtained from sensor responses on arandom line target. Next, in step S140, line centers are extracted fromthe measured sensor response. In various exemplary embodiments, thesensor response is normalized, and the line centers are extracted fromthe normalized sensor response.

Then, in step S150, sensor displacement is determined from matrixequations comparing expected and actual or measured displacement. Methodthen proceeds to step S160.

In step S160, a determination is made whether the displacementdetermination process needs to be repeated for improved accuracy. If itis determined in step S160 that improved accuracy is needed, the methodreturns to step S130. Otherwise, operation proceeds to step S170, wherethe method ends.

It should be understood that some steps in FIG. 6, such as steps S110,S120 and S160, may be omitted.

FIG. 7 is a flowchart outlining an exemplary embodiment for a lineararray sensor calibration method for the calibration-on-drum technique.This method is similar to that shown in FIG. 6, except for the featuresrelated to the calibration-on-drum technique.

As shown in FIG. 7, the method starts at step S200 and continues to stepS210, where sensor responses to the illumination level on the drum areobtained. Next, in step S220, a function is fit to the sensor responsesto extract coefficients or/and parameters. The function may be linear ornon-linear.

Then, in step S230, measurements are obtained from sensor responses at afixed illumination level as a function of rotation angle of the drum. Invarious exemplary embodiments, this step is used for drum signaturecalibration, such as drum rotational signature calibration.

Next, in step S240, a random line target is generated using x-axiscontroller to produce line pairs of known separation. Next, in stepS250, measurements are obtained from sensor response to the random linetarget.

Then, in step S260, line centers are extracted from the measured sensorresponse. In various exemplary embodiments, the measured sensor responseis normalized and the line centers are extracted from the normalizedsensor response.

Next, in step S270, sensor displacement is determined from matrixequations comparing expected and measured or actual displacement.Operation of the method proceeds to step S280.

In step S280, it is determined whether to repeat the displacementdetermination process for improved accuracy. If it is determined at S280that improved accuracy is needed, operation returns to step S240.Otherwise, operation continues to step S290, where the method ends.

It should be understood that some steps in FIG. 7, such as steps S210,S220 and S280, may be omitted.

FIG. 8 is a functional block diagram of an exemplary embodiment of alinear array sensor calibration system according to this invention. Asshown in FIG. 7, the system 300 may include an input/output interface310, a controller 320, a memory 330, a sensor response obtainingcircuit, routine or application 340, a function fitting circuit, routineor application 350, a drum signature calibrating circuit, routine orapplication 360, a line center extracting circuit, routine orapplication 370, a displacement determining circuit, routine orapplication 380, and a sensor accuracy controlling circuit, routine orapplication 390, each interconnected by one or more control and/or databuses and/or application programming interfaces 395.

In various exemplary embodiments, the system 300 is implemented on aprogrammable general purpose computer. However, the system 300 can alsobe implemented on a special purpose computer, a programmedmicroprocessor or microcontroller and peripheral integrated circuitelements, an ASIC or other integrated circuits, a digital signalprocessor (DSP), a hard wired electronic or logic circuit, such asdiscrete element circuit, a programmable logic device such as a PLD,PLA, FPGA or PAL, or the like. In general, any device capable ofimplementing a finite state machine that is in turn capable ofimplementing the flowchart shown in FIGS. 6 and 7 can be used toimplement the system 300.

The input/output interface 310 interacts with the outside of the system300. In various exemplary embodiments, the input/output interface 310may receive input from the outside, such as an input 400, via one ormore links 410. The input/output interface 310 may output data to anoutput 500 via one or more links 510.

The memory 330 may also store any data and/or program necessary forimplementing the functions of the system 300. The memory 330 can beimplemented using any appropriate combination of alterable, volatile, ornon-volatile memory or non-alterable or fixed memory. The alterablememory, whether volatile or non-volatile, can be implemented using anyone or more of static or dynamic RAM, a floppy disk and a disk drive, awritable or rewritable optical disk and disk drive, a hard drive, flashmemory or the like. Similarly, the non-alterable or fixed memory can beimplemented using any one or more of ROM, PROM, EPROM, EEPROM, anoptical ROM disk, such as a CD-ROM or a DVD-ROM disk and disk drive orthe like.

The sensor response obtaining circuit, routine or application 340obtains linear array sensor response. The function fitting circuit,routine or application 350 performs function fitting to the sensorresponse. The drum signature calibrating circuit, routine or application360 performs drum signature calibration. The line center extractingcircuit, routine or application 370 extracts line centers from thesensor response. The displacement determining circuit, routine orapplication 380 determines sensor displacement. In various exemplaryembodiments, the displacement determining circuit, routine orapplication 380 determines sensor displacement using Equation (4). Thesensor accuracy controlling circuit, routine or application 390determines whether repetition of process is necessary to improveaccuracy.

The sensor response obtaining circuit, routine or application 340, thefunction fitting circuit, routine or application 350, the drum signaturecalibrating circuit, routine or application 360, the line centerextracting circuit, routine or application 370, the displacementdetermining circuit, routine or application 380, and the sensor accuracycontrolling circuit, routine or application 390, may receive data from,or send data to, the memory 330.

In operation, the sensor response obtaining circuit, routine orapplication 340, under control of the controller 320, receives lineararray sensor response to illumination levels and sensor response torandom line target. The function fitting circuit, routine or application350, under control of the controller 320, fits a function to the sensorresponse to the illumination levels. In various exemplary embodiments,the function fitting performance establishes sensor responsecalibration.

The drum signature calibrating circuit, routine or application 360,under control of the controller 320, calibrates drum signature when thesensor has been mounted in a printer. In various exemplary embodiments,the drum signature calibrating circuit, routine or application 360, onthe controller of the controller 320, calibrates drum signature in thex-axis direction and/or drum rotational signature based on the sensorresponse to illumination levels.

The line center extracting circuit, routine or application 370, undercontrol of the controller 320, extracts line centers from the sensorresponse to the random line target. The displacement determiningcircuit, routine or application 380, under control of the controller320, determines sensor displacement based on the extracted line centersand the known spacing between the line pairs in the random line target.The sensor accuracy controlling circuit, routine or application 390,under control of the controller 320, determines whether the extractionof line centers and/or determination of sensor displacement should berepeated for improved accuracy.

While particular embodiments have been described, alternatives,modifications, variations and improvements may be implemented within thespirit and scope of the invention.

1. A method for gray level calibration of a linear array sensor using monitored reflectivity of a drum, the drum having an axis and rotating about the axis, the method comprising: determining a first set of calibration parameters when the drum is at a first rotation angle about the axis, the first set of calibration parameters associated with calibrating the linear array sensor in a direction along the axis when the drum is at the first rotation angle about the axis; determining a second set of calibration parameters when the drum is at a second rotation angle about the axis, the second set of calibration parameters associated with calibrating the linear array sensor in the direction along the axis when the drum is at the second rotation angle about the axis; and storing the first and second sets of calibration parameters.
 2. The method of claim 1, the first set of calibration parameters representing a non-linear relationship between a response of the linear array sensor to the reflectivity of the drum and the reflectivity of the drum, the method further comprising: determining the non-linear relationship based on three or more responses of the linear array sensor to three or more respective reflectivity values of the drum.
 3. The method of claim 1, further comprising: determining n^(th) set of calibration parameters when the drum is at an n^(th) rotation angle about the axis, the n^(th) set of calibration parameters associated with calibrating the linear array sensor in the direction along the axis when the drum is at the n^(th) rotation angle about the axis; and storing the n^(th) set of calibration parameters, n being an integer greater than two, and less than n=C/S, C being a circumference of the drum, and S being a distance on a surface of the drum corresponding to an angular difference between two rotation angles over which the calibration does not change significantly.
 4. A method for gray level calibration of a linear array sensor using monitored reflectivity of a belt, the belt having a length and moving in a process direction along the length, the method comprising: determining a first set of calibration parameters when the belt is at a first process direction position, the first set of calibration parameters associated with calibrating the linear array sensor in a cross process direction when the belt is at the first process direction position, the cross process direction perpendicular to the process direction; determining a second set of calibration parameters when the belt is at a second process direction position, the second set of calibration parameters associated with calibrating the linear array sensor in the cross process direction when the belt is at the second process direction position; and storing the first and second sets of calibration parameters.
 5. The method of claim 1, the first set of calibration parameters representing a non-linear relationship between a response of the linear array sensor to the reflectivity of the belt and the reflectivity of the belt, the method further comprising: determining the non-linear relationship based on three or more responses of the linear array sensor to three or more respective reflectivity values of the belt.
 6. The method of claim 4, further comprising: determining n^(th) set of calibration parameters when the belt is at an n^(th) process direction position, the n^(th) set of calibration parameters associated with calibrating the linear array sensor in the cross process direction when the belt is at the n^(th) process direction position; and storing the n^(th) set of calibration parameters, n being an integer greater than two, n=C/S, C being the length of the belt, and S being a distance on the belt between two process direction positions.
 7. A machine-readable medium that provides instructions for gray level calibration of a linear array sensor using monitored reflectivity of a drum, the drum having an axis and rotating about the axis, the instructions, when executed by a processor, causing the processor to perform operations comprising: determining a first set of calibration parameters when the drum is at a first rotation angle about the axis, the first set of calibration parameters associated with calibrating the linear array sensor in a direction along the axis when the drum is at the first rotation angle about the axis; determining a second set of calibration parameters when the drum is at a second rotation angle about the axis, the second set of calibration parameters associated with calibrating the linear array sensor in the direction along the axis when the drum is at the second rotation angle about the axis; and storing the first and second sets of calibration parameters.
 8. The machine-readable medium according to claim 7, the first set of calibration parameters representing a non-linear relationship between a response of the linear array sensor to the reflectivity of the drum and the reflectivity of the drum, the operations further comprising: determining the non-linear relationship based on three or more responses of the linear array sensor to three or more respective reflectivity values of the drum.
 9. The machine-readable medium according to claim 7, the operations further comprising: determining n^(th) set of calibration parameters when the drum is at an n^(th) rotation angle about the axis, the n^(th) set of calibration parameters associated with calibrating the linear array sensor in the direction along the axis when the drum is at the n^(th) rotation angle about the axis; and storing the n^(th) set of calibration parameters, n being an integer greater than two, and less than n=C/S, C being a circumference of the drum, and S being a distance on a surface of the drum corresponding to an angular difference between two rotation angles over which the calibration does not change significantly.
 10. A machine-readable medium that provides instructions for gray level calibration of a linear array sensor using monitored reflectivity of a belt, the belt having a length and moving in a process direction along the length, the instructions, when executed by a processor, causing the processor to perform operations comprising: determining a first set of calibration parameters when the belt is at a first process direction position, the first set of calibration parameters associated with calibrating the linear array sensor in a cross process direction when the belt is at the first process direction position, the cross process direction perpendicular to the process direction; determining a second set of calibration parameters when the belt is at a second process direction position, the second set of calibration parameters associated with calibrating the linear array sensor in the cross process direction when the belt is at the second process direction position; and storing the first and second sets of calibration parameters.
 11. The machine-readable medium according to claim 10, the first set of calibration parameters representing a non-linear relationship between a response of the linear array sensor to the reflectivity of the belt and the reflectivity of the belt, the operations further comprising: determining the non-linear relationship based on three or more responses of the linear array sensor to three or more respective reflectivity values of the belt.
 12. The machine-readable medium according to claim 10, the operations further comprising: determining n^(th) set of calibration parameters when the belt is at an n^(th) process direction position, the n^(th) set of calibration parameters associated with calibrating the linear array sensor in the cross process direction when the belt is at the n^(th) process direction position; and storing the n^(th) set of calibration parameters, n being an integer greater than two, n=C/S, C being the length of the belt, and S being a distance on the belt between two process direction positions.
 13. A system for gray level calibration of a linear array sensor using monitored reflectivity of a drum, the drum having an axis and rotating about the axis, the system comprising: a memory; a drum signature calibrating circuit, routine or application that determines a first set of calibration parameters when the drum is at a first rotation angle about the axis and a second set of calibration parameters when the drum is at a second rotation angle about the axis, the first set of calibration parameters associated with calibrating the linear array sensor in a direction along the axis when the drum is at the first rotation angle about the axis, the second set of calibration parameters associated with calibrating the linear array sensor in the direction along the axis when the drum is at the second rotation angle about the axis; and a controller that stores the first and second sets of calibration parameters in the memory.
 14. The system of claim 13, wherein: the first set of calibration parameters representing a non-linear relationship between a response of the linear array sensor to the reflectivity of the drum and the reflectivity of the drum, the drum signature calibrating circuit, routine or application determines the non-linear relationship based on three or more responses of the linear array sensor to three or more respective reflectivity values of the drum.
 15. The system of claim 13, wherein: the drum signature calibrating circuit, routine or application determines n^(th) set of calibration parameters when the drum is at an n^(th) rotation angle about the axis, the n^(th) set of calibration parameters associated with calibrating the linear array sensor in the direction along the axis when the drum is at the n^(th) rotation angle about the axis; and the controller stores the n^(th) set of calibration parameters, n being an integer greater than two, n=C/S, C being a circumference of the drum, and S being a distance on a surface of the drum corresponding to an angular difference between two rotation angles.
 16. A system for gray level calibration of a linear array sensor using monitored reflectivity of a belt, the belt having a length and moving in a process direction along the length, the system comprising: a memory; a drum signature calibrating circuit, routine or application that determines a first set of calibration parameters when the belt is at a first process direction position and a second set of calibration parameters when the belt is at a second process direction position, the first set of calibration parameters associated with calibrating the linear array sensor in a cross process direction when the belt is at the first process direction position, the cross process direction perpendicular to the process direction, the second set of calibration parameters associated with calibrating the linear array sensor in the cross process direction when the belt is at the second process direction position; and a controller that stores the first and second sets of calibration parameters.
 17. The system of claim 16, wherein: the first set of calibration parameters representing a non-linear relationship between a response of the linear array sensor to the reflectivity of the belt and the reflectivity of the belt, and the drum signature calibrating circuit, routine or application determines the non-linear relationship based on three or more responses of the linear array sensor to three or more respective reflectivity values of the belt.
 18. The system of claim 16, wherein: the drum signature calibrating circuit, routine or application determines n^(th) set of calibration parameters when the belt is at an n^(th) process direction position, the n^(th) set of calibration parameters associated with calibrating the linear array sensor in the cross process direction when the belt is at the n^(th) process direction position; and the controller stores the n^(th) set of calibration parameters, n being an integer greater than two, and less than n=C/S, C being the length of the belt, and S being a distance on the belt between two process direction positions over which the calibration does not change significantly. 